Exercises in Group Theory

1 Sets.- 1. Basic Concepts.- 2. Mappings of Sets.- 3. Binary Relations.- 4. Multiplication of Binary Relations.- 2 Algebraic Operations of a General Type.- 1. The Concept of an Algebraic Operation.- 2. Basic Properties of Operations.- 3. Multiplication of Subsets of a Multiplicative Set.- 4. Homomorphisms.- 5. Semigroups.- 6. Elementary Concepts of the Theory of Groups.- 3 Compositions of Transformations.- 1. General Properties of the Composition of Transformations.- 2. Invertible Transformations.- 3. Invertible Transformations of Finite Sets.- 4. Endomorphisms.- 5. Groups of Isometries.- 6. Partial Transformations.- 4 Groups and Their Subgroups.- 1. Decomposition of a Group by a Subgroup.- 2. Conjugate Classes.- 3. Normal Subgroups and Factor Groups.- 4. Subgroups of Finite Groups.- 5. Commutators and the Commutator Subgroup.- 6. Solvable Groups.- 7. Nilpotent Groups.- 8. Automorphisms of Groups.- 9. Transitive Groups of Transformations.- 5 Defining Sets of Relations.- 1. Defining Sets of Relations on Semigroups.- 2. Defining Sets of Relations on Groups.- 3. Free Groups.- 4. Groups Defined by Sets of Relations.- 5. Free Products of Groups.- 6. The Direct Product of Groups.- 6 Abelian Groups.- 1. Elementary Properties of Abelian Groups.- 2. Finite Abelian Groups.- 3. Finitely Generated Abelian Groups.- 4. Infinite Abelian Groups.- 7 Group Representations.- 1. Representations of a General Type.- 2. Representations of Groups by Transformations.- 3. Representations of Groups by Matrices.- 4. Groups of Homomorphisms of Abelian Groups.- 5. Characters of Groups.- 8 Topological and Ordered Groups.- 1. Metric Spaces.- 2. Groups of Continuous Transformations of a Metric Space.- 3. Topological Spaces.- 4. Topological Groups.- 5. Ordered Groups.- Hints.- 1.- 2.- 3.- 4.- 5.- 6.- 7.- 8.- Answers.- 1.- 2.- 3.- 4.- 5.- 6.- 7.- 8.